Question

A certain charge $Q$ is divided into two parts $q$ and $Q - q$. How does the charge $Q$ and $q$ must be related so that when $q$ and $Q - q$ are placed at a certain distance apart, they experience maximum electrostatic repulsion?
A) $Q = 2q$
B) $Q = 3q$
C) $Q = 4q$
D) $Q = 4q + c$

Answer 1

Hint: The relation between the $Q$ and $q$ can be determined by using the electrostatic force formula between the two charges, and then by differentiating the electrostatic force with respect to the charge, then the relation is determined.

Formula used:
The electrostatic force between the two charge is given by,
$F = \dfrac{1}{{4\pi {\varepsilon _0}}} \times \dfrac{{{q_1}{q_2}}}{{{r^2}}}$
Where, $F$ is the electrostatic force between the two charge, ${\varepsilon _0}$ is the permittivity of the free space, ${q_1}$ and ${q_2}$ are the two charge placed in some distance between them and $r$ is the distance between the two charges.

Complete step by step solution:
Given that,
The two charges are $q$ and $Q - q$,
So, ${q_1} = q$ and ${q_2} = Q - q$
Now,
The electrostatic force between the two charge is given by,
$F = \dfrac{1}{{4\pi {\varepsilon _0}}} \times \dfrac{{{q_1}{q_2}}}{{{r^2}}}\,...............\left( 1 \right)$
By substituting the two charge values in the above equation (1), then the above equation (1) is written as,
$F = \dfrac{1}{{4\pi {\varepsilon _0}}} \times \dfrac{{q\left( {Q - q} \right)}}{{{r^2}}}$
By multiplying the terms in the above equation, then the above equation is written as,
$F = \dfrac{1}{{4\pi {\varepsilon _0}}} \times \dfrac{{qQ - {q^2}}}{{{r^2}}}$
Now differentiating the above equation with respect to the charge $q$, then the above equation is written as,
$\dfrac{{\partial F}}{{\partial q}} = \dfrac{1}{{4\pi {\varepsilon _0}}} \times \dfrac{{Q - 2q}}{{{r^2}}}$
It is given that the maximum electrostatic repulsion force, so $\dfrac{{\partial F}}{{\partial q}} = 0$, then the above equation is written as,
$0 = \dfrac{1}{{4\pi {\varepsilon _0}}} \times \dfrac{{Q - 2q}}{{{r^2}}}$
By rearranging the terms in the above equation, then the above equation is written as,
$0 = \dfrac{{Q - 2q}}{{{r^2}}}$
By cross multiplying the terms in the above equation, then the above equation is written as,
$Q - 2q = 0$
By rearranging the terms in the above equation, then the above equation is written as,
$Q = 2q$

Hence, the option (A) is the correct answer.

Note: The electrostatic force is given as the repulsive electrostatic force, at the maximum repulsive force, then the force is assumed to be zero. After this assumption, then the relationship between the $Q$ and $q$ can be determined. The term $\dfrac{1}{{4\pi {\varepsilon _0}}}$ is constant.